SYNTHETIC RESEARCHES IN ALGEBRA AND ITS RELATED FIELDS

在这个研究项目中，举行了第31、32、33届代数学术会议和几次个人学术会议，为日本代数界的所有研究人员提供了各种研究方向和建议。用代数几何中的数字理论方法构造曲面的分支理论解决二维情形的Serre猜想，利用辫群构造分瓣Galois覆盖理论及其在Galois表示和Jacobi和中的应用，引入高阶圆单位的新概念，利用晶体上同调对显式互易律的有机解释及其推广到更一般交换环的可能性.在群论方面，Hall猜想、Janusz猜想作为有限单群分类定理的应用得到了解决。在组合理论中，值得注意的是， ...更多信息 给出了Iwahori代数和环上的Hadamard矩阵和几个新的多项式不变量。在代数几何中，给出了代数闭域上的单正则簇的结构理论，在交换环中引入了代数理论中的有限表示型概念，在很大程度上对Buchsbaum环进行了显式研究.最后，在Dynkin数学、几何和自同构形式讨论会上，为了使一些几何之间的研究相互交流，这些结果对统计物理、奇点、几何、对称空间数论等领域的Dynkin图的研究有着重要的意义，对代数学及其相关领域的重要课题也有一定的启发性、有机性，从而对相关领域的研究者产生了很大的影响。结果，日本代数学领域取得了广泛而显著的进展。少

IN THIS RESEARCH PROJECT THE 31TH, 32TH, 33TH SYMPOSIUMS OF ALGEBRA AND THE SEVERAL INDIVIDUAL SYMPOSIUMS WERE HELD, SO THAT THEY OFFERED VARIOUS RESEARCH DIRECTIONS, SUGGESTIONS TO ALL CORESEARCHERS WORKING IN THE ALGEBRA FIRELD OF JAPAN.SOLUTION OF SERRE'S CONJECTURE IN 2-DIMENSIONAL CASE BY CONSTRUCTING A RAMIFICATION THEORY FOR SURFACES WITH SYSTEMATIC USE OF NUMVER THEORETIC METHOD IN ALGEBRAIC GEOMETRY, CONSTRUCTION OF THE THEORY OF RAMIFIED GALOIS COVERINGS BY MAKING USE OF BRAID GROUPS AND ITS APPLICATION TO GALOIS REPRESENTATION AND JACOBI SUMS, INTRODUCTION OF A NEW CONCEPT OF HIGHER CIRCULAR UNIT, AND AN ORGANIC INTERPRETATION OF EXPLICIT RECIPROCITY LAWS THROUGH CRYSTALLINE COHOMOLOGY WITH POSSIBILITY OF ITS GENERALIZATION TO MORE GENERAL COMMUTATIVE RINGS.AS FOR THE GROUP THEORY, HALL'S CONJECTURE, JANUSZ'S CONJECTURE WERE SOLVED AS AN APPLICATION OF THE CLASSIFICATION THEOREM OF FINITE SIMPLE GROUPS. IN THE COMBINATORIAL THEORY IT IS NOTICEABLE THAT A NUMBER THEORETIC CON … More STRUCTION OF SOME HADAMARD MATRICES AND SEVERAL NEW POLYNOMIAL INVARIANTS ON IWAHORI ALGEBRA AND LINKS WERE GIVEN. IT IS ALSO WORTHWHILE THAT IN ALGEBRAIC GEOMETRY A STRUCTURE THEORY OF UNIRULED VARIETIES OVER ALGEBRAICALLY CLOSED FIELDS WAS SHOWN, THE CONCEPT OF FINITE REPRESENTATION TYPE IN ALGEBRA THEORY WAS INTRODUCED TO COMMUTATIVE RINGS, AND AN EXPLICIT STUDY ON BUCHSBAUM RING WAS DEVELOPPED TO LARGE EXTENT.FINALLY, IN THE SYMPOSIUMS ON DYNKIN MATHEMATICS, GEOMETRY AND AUTOMORPHIC FORMS IT WAS AIMED TO INTERCHANGE THEIR RESEARCHES AMONG SOME GEOMETERS, PHYSICISTS AND ALGEBRAISTS, SO THAT VARIOUS IMPORTANT RESULTS WERE OBTAINED ON DYNKIN DIAGRAMS FROM STATISTICAL PHYSICS, SINGULARLITIES, GEOMETRIES AND NUMBER THEORY OF SYMMETRIC SPACES.AS IN THE ABOVE, CERTAIN SYNTHETIC, ORGANIC DEVELOPMENTS WERE MADE ON IMPORTANT SUBJECTS NOWADAYS IN THE AREA OF ALGEBRA AND ITS RELATED FIELDS, THERETHROUGH ALL RESEARCHERS IN RELATED FIELDS WERE AFFECTED GOOD INFLUENCES, ENCOURAGEMENTS. CONSEQUENTLY, A GREAT PROGRESS IN THE ALGEBRA FIELD OF JAPAN WAS MADE WIDELY AND REMARKABLY. Less